Definition talk:Probability Generating Function

Do I have permission to rewrite this to a discussion in the setting of formal power series? All the tacit assumptions on convergence are a pain to the eye of the rigorous mathematician. --Lord_Farin 08:37, 9 June 2012 (EDT)

For example, consider $\Pr(X = n!) = 2^{-n}$ with $n \ge 1$, which' (I know that's grammatically wrong, but it's correct to logicians) power series does not converge except for $s = 0$. --Lord_Farin 11:06, 9 June 2012 (EDT)

Would it make sense to add a section "formal definition" or whatever? --prime mover 11:21, 9 June 2012 (EDT)

Making the definition rigorous would probably not imply that the discussion will be inaccessible to the uninitiated; rather, I envisage that people aren't led to believe that any distribution actually has a converging power series $\Pi$ in which arbitrary values can be plugged; rather, any distribution $X$ has an associated formal power series $\Pi_X$, by means of which useful statements can be made with regard to existence of moments (such as expected value). In conclusion, the page is up for a rewrite in any case, and I am volunteering; en passant I will make the discussion rigorous by not de facto asserting that $\Pi$ is necessarily a real analytic function (and I suspect/hope that my approach is in line with the reference). --Lord_Farin 11:55, 9 June 2012 (EDT)

Feel free. --prime mover 16:49, 9 June 2012 (EDT)

What do you say of the result? --Lord_Farin 17:36, 9 June 2012 (EDT)

For some background, the formal power series relates to ordinary power series as a polynomial relates to the function it defines. --Lord_Farin 17:40, 9 June 2012 (EDT)

I'm afraid I don't know enough about the subject to understand what was wrong with the original. --prime mover 17:56, 9 June 2012 (EDT)